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REVIEW 3 major objections 4 minor 16 references

No two Jellyfish graphs are L-cospectral and Q-cospectral

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Jellyfish graphs are claimed to be determined by their signless Laplacian spectrum, and by their Laplacian spectrum when the cycle length is even.

desk verdict Nice generalization idea, but the proofs have load-bearing errors: Lemma 3.1's bound is false for even q, and the Q-spectrum formula doesn't match the paper's own matrix. read the letter →

arxiv 1908.07909 v2 pith:TT33PPOI submitted 2019-08-19 math.CO

A. Z. Abdian , A. R. Ashrafi This is my paper · ORCID
classification math.CO MSC 05C50
keywords jellyfishgraphssunsignlessLaplacianspectrumspectraldeterminationDQSDLSunicyclic
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to prove that jellyfish graphs—graphs formed by attaching $p$ pendant leaves to each vertex of a cycle of length $q$—are determined by their spectra in two senses: no other graph can share the signless Laplacian spectrum of a jellyfish graph, and, when $q$ is even, no other graph can share its Laplacian spectrum either. These properties are called DQS and DLS, respectively. The results extend known spectral-determination theorems for sun graphs, which are the special case $p=1$. The proof strategy is to show that any cospectral candidate must have the same degree sequence as the jellyfish graph, and then to use the fact that a unicyclic graph with that degree sequence is forced to be the jellyfish graph itself.

What carries the argument

The load-bearing object is the degree sequence $(1^{pq}, (p+2)^q)$ of $JFG(p,q)$—the $q$ cycle vertices each carry degree $p+2$, and the $pq$ leaves have degree 1. The proofs force any cospectral candidate to reproduce this sequence by combining spectral moments of the Laplacian or signless Laplacian with counts of closed walks and triangles in the line graph, using the spanning-tree count to fix the cycle length. On the Laplacian side, the cap $\mu_1(JFG(p,q)) \leq p+3+\frac{2}{p+2}$ is what limits the maximum degree of a cospectral graph to $p+2$; on the Q-side, an explicit block-matrix formula gives the eigenvalues $\lambda_i + p+3 \pm \frac{1}{2}\sqrt{\lambda_i^2+(2p+2)\lambda_i+p^2+2p+5}$ with $\lambda_i=\cos(2\pi i/q)$, plus a multiplicity of 1.

What would settle it

Compute the Laplacian spectrum of the 8-vertex graph $JFG(1,4)$, a 4-cycle with one pendant leaf at each vertex: its largest Laplacian eigenvalue is $3+\sqrt5 \approx 5.236$, which exceeds the claimed upper bound $14/3 \approx 4.667$ in Lemma 3.1. This falsifies the spectral-radius bound behind the even-$q$ degree-sequence argument; a search for other graphs with the same degree sequence and spanning-tree count as $JFG(p,q)$ for small even $q$ would settle whether the DLS conclusion itself survives.

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Extended reading notes

Core claim

The paper's central claim is that the jellyfish graph $JFG(p,q)$ is DQS for all $p,q$, and DLS when $q$ is even. For the signless Laplacian side, the paper derives the full spectrum of any Q-cospectral graph from a block decomposition of the signless Laplacian, then uses spectral-moment identities to recover the degree sequence and connectedness to force isomorphism. For the Laplacian side, it argues that any L-cospectral graph has the same order, size, number of spanning trees, and first Zagreb index as the jellyfish graph; under a bound on the largest Laplacian eigenvalue, these force the same degree sequence, and the spanning-tree count fixes the cycle length, so the candidate must be the jellyfish graph. A corollary transfers the Laplacian determination to complements of such graphs.

Load-bearing premise

The Laplacian-side proof rests on the bound $\mu_1(JFG(p,q)) \le p+3+\frac{2}{p+2}$, used to conclude that any L-cospectral graph has maximum degree at most $p+2$; for even $q$ the bound is asserted but not generally true, since $JFG(1,4)$ has Laplacian spectral radius $3+\sqrt5 \approx 5.236$, larger than $14/3 \approx 4.667$.

Editorial extensions

If this is right

  • Every jellyfish graph $JFG(p,q)$ is claimed to be uniquely determined by its signless Laplacian spectrum, so any graph with the same $Q$-spectrum must be isomorphic to it.
  • For even $q$, any graph with the same Laplacian spectrum as a jellyfish graph is claimed to be isomorphic to it, and so is the complement of that graph.
  • A graph Q-cospectral with $JFG(p,q)$ must have degree sequence $(1^{pq}, (p+2)^q)$, according to the proof.
  • The explicit $Q$-spectrum formula gives the spectral radius $\frac{p+5+\sqrt{p^2+6p+13}}{2}$ for any jellyfish graph and for anything cospectral with it.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the even-$q$ Laplacian bound fails generally, the DLS conclusion may still hold, but the proof would need a different argument; the natural next target is the correct Laplacian spectral radius bound for $JFG(p,q)$.
  • The conjecture that odd-$q$ jellyfish graphs are also DLS could be tested with the same degree-sequence equations, which do not depend on the disputed bound on the Q-side.
  • Because the mechanism is essentially degree-sequence forcing, the technique may extend to other unicyclic graphs built by coalescing stars with cycles, a direction the paper does not explore.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript claims that jellyfish graphs JFG(p,q), obtained by attaching p leaves to each vertex of a q-cycle, are determined by their signless Laplacian spectrum (DQS) and, when q is even, by their Laplacian spectrum (DLS). The proof strategy for the DLS half is to show via a spectral-radius bound that any L-cospectral graph has the same degree sequence and is unicyclic, hence isomorphic; the DQS half proceeds by computing the Q-spectrum explicitly, then using spectral moments and a connectivity argument to force the degree sequence and the graph structure. The paper also poses a conjecture for odd q in the Laplacian case.

Significance. The topic is appropriate for a spectral graph theory journal: extending determinability results from sun graphs to a natural family is of interest, and the paper includes explicit formulas (the Q-spectrum expression in Lemma 3.3 and the spectral-moment equations in Lemma 3.5) that are potentially useful. However, the central technical lemmas contain errors that invalidate both main theorems. Lemma 3.1's spectral radius bound is demonstrably false for even q, and Lemma 3.3's Q-spectrum computation is algebraically inconsistent with its own block matrix. These are not superficial typos: the subsequent degree-sequence arguments and connectivity proof depend directly on these claims. As a result, the manuscript does not establish either DQS or DLS for jellyfish graphs.

major comments (3)
  1. [Lemma 3.1] The claimed upper bound μ1(JFG(p,q)) ≤ p+3+2/(p+2) is false for even q. For JFG(2,4), the alternating cycle mode x_i = (-1)^i with leaf values y_i = -x_i/(λ-1) gives Laplacian eigenvalue λ satisfying (λ-p-4)(λ-1)=p, i.e., λ² - 7λ + 4 = 0, so μ1 = (7+√33)/2 ≈ 6.372, exceeding the claimed bound 5.5. Since Lemma 3.2 uses this bound together with Theorem 2.3 to conclude d1(H) ≤ p+2, Eq. (4) does not follow and the degree-sequence argument in Lemma 3.2 collapses. This invalidates Theorem 3.1, which is the paper's DLS claim for even q.
  2. [Lemma 3.3] The displayed block matrix has B and C blocks consisting of p copies of I_q stacked vertically, so the Schur complement of D = I_{n-q} in the characteristic determinant is p/(x-1), not 1/(x-1). Consequently the expression for the Q-spectrum is valid only for p=1. Corollary 3.2 and the subsequent use of the spectral radius in Lemma 3.6 rely on this formula, so the DQS proof is not established for p ≥ 2.
  3. [Lemma 3.6] In case (2a), the proof asserts that 'by Lemma 3.5, there exists a subgraph G1 of H_j such that G1 ≅ JFG(p,q')' for some unspecified q'. Lemma 3.5 establishes only that H and G have the same degree sequence; it does not imply the existence of a jellyfish subgraph with the same p and some cycle length. This unsupported assertion is load-bearing for the connectivity argument, and the same gap reappears in case (2b).
minor comments (4)
  1. [Lemma 3.1] The proof cites Lemmas 2.3 and 2.4 for the two spectral-radius inequalities, but the statements of those lemmas do not directly supply the bounds; the citations appear to be mismatched.
  2. [Lemma 3.2] The notation n'_{p+2} for the count of degree-(p+2) vertices in G is introduced with a prime that carries no meaning and could be confused with a complementary quantity; please remove the prime.
  3. [Lemma 3.4] The text 'det(Q(H) = 4' is missing a closing parenthesis before '= 4'.
  4. [Theorem 3.2] The step 'H is an unicyclic graph and so H = G' needs justification: the degree sequence alone does not immediately force H to be unicyclic, and the citation to Lemma 3.6 should be made explicit here.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation relies on standard external spectral results, and the only self-citation is not load-bearing.

full rationale

The paper's derivation chain is a spectral characterization argument: it defines jellyfish graphs structurally, then uses standard spectral tools (Laplacian/signless Laplacian spectral moments, closed-walk identities, line-graph cospectrality, the Grone-Merris bound) to force the degree sequence and unicyclic structure from cospectrality. No parameter is fitted, no quantity is defined in terms of the target conclusion, and no 'prediction' is renamed from an input. The only self-citation is reference [1], paired with the independent standard reference [6] for the theorem μ1(G) ≥ d1(G)+1; that bound is not special to this paper and does not import the main result. Alleged flaws such as the false bound in Lemma 3.1 for even q are correctness concerns, not circularity: an invalid proof step does not make the argument equivalent to its inputs. The paper is therefore not circular in the sense of the requested analysis.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The proofs use standard theorems from the cited literature as axioms: spectral moment formulas, Laplacian eigenvalue bounds, line graph cospectrality, and properties of the Q-spectrum. No free parameters are fitted and no new entities are introduced. The paper does not prove these background results, which is normal for this area.

assumptions (5)
  • standard math Laplacian spectrum determines n, m, number of components, spanning trees, and first Zagreb index
    Invoked in Lemmas 3.2 and 3.5 without proof; this is standard spectral graph theory.
  • standard math Spectral moment formulas for Q-spectrum (Lemma 2.1): T0=n, T1=2m, T2=2m+Σd_i^2, T3=6Δ+3Σd_i^2+Σd_i^3
    Used to write equations (6)-(8) in Lemma 3.5.
  • standard math If G and H are Q-cospectral then their line graphs are A-cospectral (Lemma 2.4(2))
    Basis for transferring triangle counts from L(G) to L(H) in Lemma 3.5.
  • standard math Laplacian spectral radius bounds: μ1 ≥ d1+1 and μ1 ≤ max_v(deg(v)+θ(v)) (Theorems 2.3, 2.4)
    Used in Lemma 3.1 to bound μ1(G) and d1(H). The claimed derived bound is false for even q.
  • standard math A proper subgraph of a connected graph has strictly smaller Q-spectral radius (Lemma 2.5)
    Used in Lemma 3.6 to derive contradiction from a supposed jellyfish subgraph.

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Cite this review

Pith. "Pith review of No two Jellyfish graphs are L-cospectral and Q-cospectral." pith.science (2026). https://pith.science/paper/TT33PPOI

@misc{pith2026190807909,
  author       = {Pith},
  title        = {Pith review of: No two Jellyfish graphs are L-cospectral and Q-cospectral},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TT33PPOI}},
  note         = {Machine review of arXiv:1908.07909}
}
read the original abstract

In this paper, it is proved that the jellyfish graphs, a natural generalization of sun graphs, are both DLS and DQS.

Figures

Figures reproduced from arXiv: 1908.07909 by the authors.

Figure 1
Figure 1. The jellyfish graph JF G(p, q) t(L(H)) = t(L(G)) = d X1(H) i=0 ni  i 3  = q  p + 2 3  (11) , 2m(L(H)) = 2m(L(G)) = d X1(H) i=0 2ni  i 2  = 2q  p + 2 2  (12) . We claim that d1(H) = p + 2. Suppose on the contrary that d1(H) ≤ p + 1 or d1(H) ≥ p + 3. We consider the following two cases: (1) d1(H) ≤ p + 1. Then by (11), q [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗

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Reference graph

Works this paper leans on

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